Graphs

A graph is simply a collection of vertices (or nodes) and edges (or ties).
We can denote this $\mathcal{G}(V,E)$, where $V$ is a the vertex set and $E$ is the edge set.
The vertices of the graph represent the actors in the social system. These are usually individual people, but they could be households, geographical localities, institutions, or other social entities.
The edges of the graph represent the relations between these entities (e.g., “is friends with” or “has sexual intercourse with” or “sends money to”). These edges can be directed (e.g., “sends money to”) or undirected (e.g., “within 2 meters of”).
When the relations that define the graph are directional, we have a directed graph or digraph.
Graphs (and digraphs) can be binary (i.e., presence/absence of a relationship) or valued (e.g., “groomed five times in the observation period”, “sent $100”).
A graph (with no self-loops) with $n$ vertices has ${n \choose 2} = n(n-1)/2$ possible unordered pairs. This number (which can get very big!) is important for defining the density of a graph, i.e., the fraction of all possible relations that actually exist in a network.

Some Definitions

Graph: Collection of vertices (or nodes) and undirected edges (or ties), denoted $\mathcal{G}(V,E)$, where $V$ is a the vertex set and $E$ is the edge set.
Digraph (Directed Graph): Collection of vertices (or nodes) and directed edges.
Bipartite Graph: Graph where all the nodes of a graph can be partitioned into two sets $\mathcal{V}_1$ and $\mathcal{V}_2$ such that for all edges in the graph connects and unordered pair where one vertex comes from $\mathcal{V}_1$ and the other from $\mathcal{V}_2$. Often called an “affiliation graph” as bipartite graphs are used to represent people’s affiliations to organizations or events.

Various Ways to Specify Graphs

igraph is a package that provides tools for the analysis and visualization of networks
Create a small, undirected graph of five vertices from a vector of vertex pairs

require(igraph)
g <- make_graph( c(1,2, 1,3, 2,3, 2,4, 3,5, 4,5), n=5, dir=FALSE )
plot(g, vertex.color="skyblue2")

Create a small graph using graph_from_literal()
Undirected edges are indicated with one or more dashes -, --, etc. It doesn’t matter how many dashes you use – you can use as many as you want to make your code more readable.
The colon operator : links “vertex sets” – i.e., creates ties between all members of two groups of vertices

g <- graph_from_literal(Fred-Daphne:Velma-Shaggy, Fred-Shaggy-Scooby)
plot(g, vertex.shape="none", vertex.label.color="black")

Make directed edges using -+ where the plus indicates the direction of the arrow, i.e., A --+ B creates a directed edge from A to B
A mutual edge can be created using +-+

Special Graphs: Empty, Full, Ring

I really don’t like the current default color in igraph (a kind of sickly orange), so I set the vertex color for every plot – you don’t have to do that

# empty graph
g0 <- make_empty_graph(20)
plot(g0, vertex.color="skyblue2", vertex.size=10, vertex.label=NA)

# full graph
g1 <- make_full_graph(20)
plot(g1, vertex.color="skyblue2", vertex.size=10, vertex.label=NA)

# ring
g2 <- make_ring(20)
plot(g2, vertex.color="skyblue2", vertex.size=10, vertex.label=NA)

Special Graphs: Lattice, Tree, Star

# lattice
g3 <- make_lattice(dimvector=c(10,10))
plot(g3, vertex.color="skyblue2", vertex.size=10, vertex.label=NA)

# tree
g4 <- make_tree(20, children = 3, mode = "undirected")
plot(g4, vertex.color="skyblue2", vertex.size=10, vertex.label=NA)

# star
g5 <- make_star(20, mode="undirected")
plot(g5, vertex.color="skyblue2", vertex.size=10, vertex.label=NA)

Special Graphs: Erdos-Renyi & Power-Law

# Erdos-Renyi Random Graph
g6 <- sample_gnm(n=100,m=50)
plot(g6, vertex.color="skyblue2", vertex.size=5, vertex.label=NA)

# Power Law
g7 <- sample_pa(n=100, power=1.5, m=1,  directed=FALSE)
plot(g7, vertex.color="skyblue2", vertex.size=5, vertex.label=NA)

Putting Graphs Together

Sometimes you want to plot two (or more) graphs together
The disjoint union operator allows you to merge two graphs with different vertex sets

plot(g4 %du% g7, vertex.color="skyblue2", vertex.size=5, vertex.label=NA)

Rewiring

gg <- g4 %du% g7
gg <- rewire(gg, each_edge(prob = 0.3))
plot(gg, vertex.color="skyblue2", vertex.size=5, vertex.label=NA)

## retain only the connected component
gg <- induced.subgraph(gg, subcomponent(gg,1))
plot(gg, vertex.color="skyblue2", vertex.size=5, vertex.label=NA)

Vertex and Edge Attributes

You can add arbitrary attributes to both vertices and edges. Generally, you do this to store information for plotting: colors, edge weights, names, etc.
Some attributes are automatically created when you construct an graph object (e.g., “name” or “weight” if you load a weighted adjacency matrix)
V(g) accesses vertex attributes
E(g) accesses edge attributes

## look at the structure
g4

## IGRAPH f9fc1e1 U--- 20 19 -- Tree
## + attr: name (g/c), children (g/n), mode (g/c)
## + edges from f9fc1e1:
##  [1] 1-- 2 1-- 3 1-- 4 2-- 5 2-- 6 2-- 7 3-- 8 3-- 9 3--10 4--11 4--12 4--13
## [13] 5--14 5--15 5--16 6--17 6--18 6--19 7--20

V(g4)$name <- LETTERS[1:20]
## see how it's changed
g4

## IGRAPH f9fc1e1 UN-- 20 19 -- Tree
## + attr: name (g/c), children (g/n), mode (g/c), name (v/c)
## + edges from f9fc1e1 (vertex names):
##  [1] A--B A--C A--D B--E B--F B--G C--H C--I C--J D--K D--L D--M E--N E--O E--P
## [16] F--Q F--R F--S G--T

## see what I did there?
## hexadecimal color codes
V(g4)$vertex.color <- "#4503fc"
E(g4)$edge.color <- "#abed8e"
g4

## IGRAPH f9fc1e1 UN-- 20 19 -- Tree
## + attr: name (g/c), children (g/n), mode (g/c), name (v/c),
## | vertex.color (v/c), edge.color (e/c)
## + edges from f9fc1e1 (vertex names):
##  [1] A--B A--C A--D B--E B--F B--G C--H C--I C--J D--K D--L D--M E--N E--O E--P
## [16] F--Q F--R F--S G--T

plot(g4, vertex.size=15, vertex.label=NA, vertex.color=V(g4)$vertex.color, 
     vertex.frame.color=V(g4)$vertex.color,
     edge.color=E(g4)$edge.color, edge.width=3)

Adjacency Matrices

Most primatologists/behavioral ecologists probably have experience thinking in terms of adjacency matrices
An example of an adjacency matrix is the pairwise interaction matrices (e.g., agonistic or affiliative interactions) that we construct from behavioral observations
A very important potential gotcha: when you read data into R, it will be in the form of a data frame. Converting an adjacency matrix to an igraph graph object requires the data to be in the matrix class. Therefore, you need to coerce the data you read in by wrapping your read.table() in an as.matrix() command.

kids <- as.matrix(
  read.table("data/strayer_strayer1976-fig2.txt",
                             header=FALSE)
  )
kid.names <- c("Ro","Ss","Br","If","Td","Sd","Pe","Ir","Cs","Ka",
                "Ch","Ty","Gl","Sa", "Me","Ju","Sh")
colnames(kids) <- kid.names
rownames(kids) <- kid.names
g <- graph_from_adjacency_matrix(kids, mode="directed", weighted=TRUE)
lay <- layout_with_fr(g)
plot(g, layout=lay, edge.arrow.size=0.5,
     vertex.color="skyblue2", vertex.label.family="Helvetica", 
     vertex.frame.color="skyblue2")

Note that you might want to change some of the graphics parameters depending on the type of display you use. For this document, the figures are constrained to be small, so you don’t want edges – and particularly arrows – to be too thick.

Adjacency matrices are actually very inefficient
- Most sociomatrices are quite sparse
- Cost of an adjacency matrix increases as $k^2$
Edge Lists are much more efficient

Community Structure

Various algorithms for detecting clusters of similar vertices – i.e., “communities”
Use fastgreedy.community() to identify clusters in Kapferer’s tailor shop and color the vertices based on their membership
fastgreedy.community() identifies four clusters
These clusters are listed as numbers in fg$membership
Use this vector to index vertex colors

A <- as.matrix(
  read.table(file="data/kapferer-tailorshop1.txt", 
             header=TRUE, row.names=1)
  )
G <- graph.adjacency(A, mode="undirected", diag=FALSE)
fg <- fastgreedy.community(G)
cols <- c("blue","red","black","magenta")
plot(G, vertex.shape="none",
     vertex.label.cex=0.75, edge.color=grey(0.85), 
     edge.width=1, vertex.label.color=cols[fg$membership],
     vertex.label.family="Helvetica")

# another approach to visualizing
plot(fg,G,vertex.label=NA)

Laying Out Graphs “By Hand”

The layout is of any given plot is random (e.g., plot the same graph repeatedly and you’ll see that the layout changes with each plot)
igraph provides a tool for tinkering with the layout called tkplot()
Call tkplot() and it will open an X11 window (on Macs at least)
Select and drag the vertices into the layout you want, then use tkplot.getcoords(gid) to get the coordinates (where gid is the graph id returned when calling tkplot() on your graph)

tkplot() window of triangle graph

g <- graph( c(1,2, 2,3, 1,3), n=3, dir=FALSE)
plot(g)

#tkplot(g)
#tkplot.getcoords(1)
### do some stuff with tkplot() and get coords which we call tri.coords
## tkplot(g)
## tkplot.getcoords(1) ## the plot id may be different depending on how many times you've called tkplot()
##     [,1] [,2]
##[1,]  228  416
##[2,]  436    0
##[3,]   20    0
tri.coords <- matrix( c(228,416, 436,0, 20,0), nr=3, nc=2, byrow=TRUE)
par(mfrow=c(1,2))
plot(g, vertex.color="skyblue2",
     vertex.frame.color="skyblue2", 
     vertex.label.family="Helvetica")
plot(g, layout=tri.coords, 
     vertex.color="skyblue2", 
     vertex.frame.color="skyblue2", vertex.label.family="Helvetica")

You may have noticed that the lattice we plotted when we introduced make_lattice() was a bit funky. This is because for a force-based layout, vertices on the periphery will have very different forces working on them than those in the center.
To get a proper lattice layout, specify that you want it on a grid

plot(g3, vertex.color="skyblue2", 
     layout=layout_on_grid(g3,10,10), vertex.size=10, vertex.label=NA)

Plotting Affiliation Graphs

davismat <- as.matrix(
  read.table(file="data/davismat.txt", 
            row.names=1, header=TRUE)
  )
southern <- graph_from_incidence_matrix(davismat) 
V(southern)$shape <- c(rep("circle",18), rep("square",14))
V(southern)$color <- c(rep("blue",18), rep("red", 14))
plot(southern, layout=layout.bipartite)

## not so beautiful
## did some tinkering using tkplot()...
x <- c(rep(23,18), rep(433,14))
y <- c(44.32432,   0.00000, 132.97297,  77.56757,  22.16216, 110.81081, 155.13514,
       199.45946, 177.29730, 243.78378, 332.43243, 410.00000, 387.83784, 354.59459,
       310.27027, 221.62162, 265.94595, 288.10811,   0.00000,  22.16216,  44.32432,
       66.48649,  88.64865, 132.97297, 166.21622, 199.45946, 277.02703, 365.67568,
       310.27027, 343.51351, 387.83784, 410.00000)
southern.layout <- cbind(x,y)
plot(southern, layout=southern.layout, vertex.label.family="Helvetica")

The incidence matrix is $n \times k$, where $n$ is the number of actors and $k$ is the number of events
Project the incidence matrix $X$ into social space, creating a sociomatrix $A$, $\mathbf{A} = \mathbf{X}\, \mathbf{X}^T$
This transforms the $n \times k$ into an $n \times n$ sociomatrix

#Sociomatrix
(f2f <- davismat %*% t(davismat))

##           EVELYN LAURA THERESA BRENDA CHARLOTTE FRANCES ELEANOR PEARL RUTH
## EVELYN         8     6       7      6         3       4       3     3    3
## LAURA          6     7       6      6         3       4       4     2    3
## THERESA        7     6       8      6         4       4       4     3    4
## BRENDA         6     6       6      7         4       4       4     2    3
## CHARLOTTE      3     3       4      4         4       2       2     0    2
## FRANCES        4     4       4      4         2       4       3     2    2
## ELEANOR        3     4       4      4         2       3       4     2    3
## PEARL          3     2       3      2         0       2       2     3    2
## RUTH           3     3       4      3         2       2       3     2    4
## VERNE          2     2       3      2         1       1       2     2    3
## MYRNA          2     1       2      1         0       1       1     2    2
## KATHERINE      2     1       2      1         0       1       1     2    2
## SYLVIA         2     2       3      2         1       1       2     2    3
## NORA           2     2       3      2         1       1       2     2    2
## HELEN          1     2       2      2         1       1       2     1    2
## DOROTHY        2     1       2      1         0       1       1     2    2
## OLIVIA         1     0       1      0         0       0       0     1    1
## FLORA          1     0       1      0         0       0       0     1    1
##           VERNE MYRNA KATHERINE SYLVIA NORA HELEN DOROTHY OLIVIA FLORA
## EVELYN        2     2         2      2    2     1       2      1     1
## LAURA         2     1         1      2    2     2       1      0     0
## THERESA       3     2         2      3    3     2       2      1     1
## BRENDA        2     1         1      2    2     2       1      0     0
## CHARLOTTE     1     0         0      1    1     1       0      0     0
## FRANCES       1     1         1      1    1     1       1      0     0
## ELEANOR       2     1         1      2    2     2       1      0     0
## PEARL         2     2         2      2    2     1       2      1     1
## RUTH          3     2         2      3    2     2       2      1     1
## VERNE         4     3         3      4    3     3       2      1     1
## MYRNA         3     4         4      4    3     3       2      1     1
## KATHERINE     3     4         6      6    5     3       2      1     1
## SYLVIA        4     4         6      7    6     4       2      1     1
## NORA          3     3         5      6    8     4       1      2     2
## HELEN         3     3         3      4    4     5       1      1     1
## DOROTHY       2     2         2      2    1     1       2      1     1
## OLIVIA        1     1         1      1    2     1       1      2     2
## FLORA         1     1         1      1    2     1       1      2     2

gf2f <- graph_from_adjacency_matrix(f2f, mode="undirected", diag=FALSE, add.rownames=TRUE)
gf2f <- simplify(gf2f)
plot(gf2f, vertex.color="skyblue2",vertex.label.family="Helvetica")

## who is the most central?
cb <- betweenness(gf2f)
#plot(gf2f,vertex.size=cb*10, vertex.color="skyblue2")
plot(gf2f,vertex.label.cex=1+cb/2, vertex.shape="none",vertex.label.family="Helvetica")

Project the matrix into event space

### this gives you the number of women at each event (diagonal) or mutually at 2 events
(e2e <- t(davismat) %*% davismat)

##     E1 E2 E3 E4 E5 E6 E7 E8 E9 E10 E11 E12 E13 E14
## E1   3  2  3  2  3  3  2  3  1   0   0   0   0   0
## E2   2  3  3  2  3  3  2  3  2   0   0   0   0   0
## E3   3  3  6  4  6  5  4  5  2   0   0   0   0   0
## E4   2  2  4  4  4  3  3  3  2   0   0   0   0   0
## E5   3  3  6  4  8  6  6  7  3   0   0   0   0   0
## E6   3  3  5  3  6  8  5  7  4   1   1   1   1   1
## E7   2  2  4  3  6  5 10  8  5   3   2   4   2   2
## E8   3  3  5  3  7  7  8 14  9   4   1   5   2   2
## E9   1  2  2  2  3  4  5  9 12   4   3   5   3   3
## E10  0  0  0  0  0  1  3  4  4   5   2   5   3   3
## E11  0  0  0  0  0  1  2  1  3   2   4   2   1   1
## E12  0  0  0  0  0  1  4  5  5   5   2   6   3   3
## E13  0  0  0  0  0  1  2  2  3   3   1   3   3   3
## E14  0  0  0  0  0  1  2  2  3   3   1   3   3   3

ge2e <- graph_from_adjacency_matrix(e2e, mode="undirected", diag=FALSE, add.rownames=TRUE)
ge2e <- simplify(ge2e)
plot(ge2e, vertex.size=20, vertex.color="skyblue2",vertex.label.family="Helvetica")

Introduction to iGraph

James Holland Jones

04/09/2023